The bootstrap backed by a proof discards the calendar window. Under a misspecified exponential likelihood, it raises coverage of a nominal 95% interval from 0.58 to 0.92.
An ACD model applies a GARCH-shaped recursion to the gaps between consecutive trades. Durations satisfy x_i = ψ_i ε_i. In the first-order version, ψ_i = ω + αx_{i-1}; the ACD(1,1) commonly estimated in practice uses ψ_i = ω + αx_{i-1} + βψ_{i-1}. Exponential quasi-likelihood fitted to tick timestamps measures how strongly a long gap predicts further long gaps. Persistence is α + β, with α + β = 1 marking the integrated boundary, the duration counterpart of IGARCH.
Cavaliere, Mikosch, Rahbek and Vilandt expose what the GARCH analogy obscures. Choose a clock window [0, T], then collect every arrival inside it: n(T) = max{k: Σx_i ≤ T}. The process under estimation therefore determines the event count. If the stationary duration distribution has tail index κ > 1, its mean is finite and the MLE converges at rate √T. At κ = 1, the mean is infinite and the rate falls to √(T/log T). Below one, convergence occurs at √(T^κ), with a mixed normal limit because the count approaches a random limit built from a κ-stable variable rather than a constant. Three regimes bring three rates. Yet the studentized statistic is N(0,1) throughout, leaving the practitioner free from identifying the regime first.
The paper asks which feature a bootstrap must reproduce.
What should remain fixed?
Two schemes compete. The random-count version regenerates durations recursively until their cumulative sum reaches the original span T. Its bootstrap count n(T) remains random while the calendar window stays fixed. That construction follows the data collection mechanism. The fixed-count version instead imposes n = n(T), allowing the bootstrap span T = Σx_i to move independently of T. It matches the realized event count while missing the clock. For point-process data, the choice appears backwards. It is nevertheless the usual approach in the ACD and multiplicative error model bootstrap literature.
Theory favors fixed-count. For κ ≥ 1, the scheme consistently estimates the estimator's law. For 0 < κ < 1, the bootstrap limit measure is random, and the scheme captures only the conditional Gaussian component of the mixed normal. The abstract acknowledges the failure directly, then gives the more useful result in the same sentence. Percentile and reverse-percentile inference remain valid, the bootstrap p-value is asymptotically uniform, and the bootstrap t-statistic is asymptotically standard normal for every κ > 0. Textbook consistency fails while first-order valid intervals survive. With exponential innovations, fixed-count bootstrap-t coverage ranges from 0.92 to 0.93 across all three κ_0 values, showing that first-order validity need not deliver 0.95 in finite samples.
The conclusion accepts the argument's central difficulty. Standard deterministic-sample-size bootstrap reasoning cannot establish fixed-count validity because n(T) is itself calculated from the durations. Auxiliary renewal lemmas carry the likelihood expansions across that random index. The random-count scheme matches the sampling mechanism, yet receives no validity theory here and appears only in simulations. Its parametric form coincides with the point-process bootstrap of Cavaliere, Lu, Rahbek and Staerk-Ostergaard (2023), whose theory requires E[x_i] to be finite. Existing work therefore covers random-count inference in the finite-mean case, while the infinite-mean case remains uncovered.
Misspecification is where the gain appears
The simulations use 10,000 replications and 399 bootstrap draws per replication, with ω_0 = 1. The calibration sets α_0 for κ ∈ {0.5, 1.0, 1.1}, producing median event counts from 200 to 1600. Innovations follow either an exponential distribution or Lomax, the heavy-tailed alternative, with shape 2.1, 3 or ∞. Their variances are 21, 3 and 1.
When innovations are exponential, the conventional studentized interval achieves 0.95 to 0.96 coverage at a 95% nominal level. Fixed-count bootstrap-t coverage reaches 0.92 to 0.93 and changes little as the sample grows from 200 events to 1600. Bootstrapping costs a little when the likelihood is correctly specified.
Misspecification reverses the ranking. At Lomax shape 3, asymptotic interval coverage drops to 0.76 to 0.82. Shape 2.1 drives it down further, to 0.58 to 0.68. Fixed-count coverage remains at 0.86 to 0.94 for the basic percentile interval and 0.89 to 0.92 for bootstrap-t. Test size makes the failure clearer. A nominal 5% two-sided test on α rejects at 0.18 to 0.24 under shape 3, then at 0.32 to 0.42 under shape 2.1. Distortion increases with sample size: for κ = 1.1, rejection rises from 0.34 at 200 events to 0.42 at 1600. Both restricted bootstrap tests bring rejection frequencies back to 0.05 to 0.09 in every design. The paper describes that as close to nominal; a small upward bias remains.
Random-count basic percentile intervals are wider throughout. With κ = 1.1, exponential innovations and 200 events, average length is 0.72, compared with 0.53 for fixed-count percentile intervals. At κ = 0.5 and shape 2.1, the comparison is 2.45 against 1.48. Bootstrap-t reverses the length ordering. For κ = 1.1, shape 2.1 and 200 events, random-count bootstrap-t has average length 1.40 and coverage 0.85. Fixed-count records 1.65 and 0.90. The shorter interval is less accurate.
Five crypto ETFs beyond the boundary
The application draws on NASDAQ LOBSTER order-book data for BTC, ETH, GBTC, ETHE and BITB. It covers 35 trading days from January 2, 2025, during regular hours only, spanning 819,000 raw seconds. Counts range from 19,366 durations for BTC to 157,620 for GBTC. Every estimate of α + β exceeds one, running from 1.002 for BITB to 1.018 for ETH. The null of infinite expected durations therefore survives against the finite-mean alternative α + β < 1.
Critical values from the bootstrap bear little resemblance to ±1.96. Restricted-residual fixed-count quantiles are [−4.05, 4.05] for BTC and [−7.36, 6.54] for ETHE. The observed t-statistics are 9.11, 23.24, 35.88, 33.23 and 3.11. Four names reject integration. BITB does not: its 3.11 falls within [−4.66, 4.02], although a normal critical value would declare it comfortably significant. Fixed-count and random-count inference also diverge visibly. For ETH, the α interval is [0.111, 0.135] under fixed-count and [0.047, 0.171] under random-count.
An upward rejection of α + β = 1 places these series in the κ < 1 regime. Classical bootstrap consistency fails there, leaving only the conditional-Gaussian argument. The empirical findings consequently rest on the weaker theoretical leg. We did not find a κ estimate anywhere in the application; the regime is inferred from the sign of α̂ + β̂ − 1.
Two further qualifications matter. The proof covers only ψ_i = ω + αx_{i-1}. Remark 2.1 says this specification keeps the bootstrap arguments transparent and frames the results as the fixed-count counterpart to non-bootstrap asymptotic theory in the analytically simplest case. The paper gives no proof for the ACD(1,1) estimated in the application.
The bootstrapped durations are also deseasonalized using cubic splines with knots every 30 minutes, an adjustment the paper calls standard practice. The paper does not say that the spline fit is held out of sample. According to the procedure as described, resampled durations therefore contain sample-wide information about the intraday shape. With 399 replications, the reported 2.5% and 97.5% quantiles depend on roughly ten tail draws apiece. BITB's borderline result is more exposed to that issue than GBTC's 35.88.
We could not run this ourselves. Our data consists of daily and one-minute bars for US equities, ETFs and crypto, whereas the required object is the number of seconds between consecutive trade prints. One-minute bars cannot recover an irregularly spaced event sequence, much less the random event count on which the argument depends.
A κ estimate beside the persistence figures would change my reading. Fixed-count bootstrap inference is consistent at κ ≥ 1. For 0 < κ < 1, it retains only first-order validity through the conditional Gaussian component. The applicable case depends on which side of one contains the tail index. For BITB, α̂ + β̂ = 1.002 is a thin basis for deciding.